TOWARDS DEEP AND SIMPLE UNDERSTANDING OF THE TRANSCENDENTAL EIGENPROBLEM OF STRUCTURAL VIBRATIONS

When using exact methods for undamped free vibration problems the generalized linear eigenvalue problem (K−ω2M) D=0 of approximate methods, e.g., finite elements, is replaced by the transcendental eigenvalue problem K (ω) D=0. Here ω is the circular frequency; D is the displacement amplitude vector; M and K are the mass and static stiffness matrices; and K (ω) is the dynamic stiffness matrix, with coefficients which include trigonometric and hyperbolic functions involving ω and mass because elements (for example, bars or beams) are analyzed exactly by solving their governing differential equations. The natural frequencies of this transcendental eigenvalue problem are generally found by the Wittrick–Williams algorithm which gives the number of natural frequencies below ωt, a trial value of ω, as ∑Jm+s{K (ωt)} wheres {} denotes the readily computed sign count property of K (ω) and the summation is over the clamped–clamped natural frequencies of all elements of the structure. Understanding the alternative solution forms of the transcendental eigenvalue problem is important both to accelerate convergence to natural frequencies, e.g., by plotting ∣K (ω)∣, and to improve the mode calculations, which lack the complete reliability of natural frequencies obtained by using the Wittrick–Williams algorithm. The three solution forms are: ∣K (ω)∣=0; D=0 with ∣K (ω)∣→∞; and ∣K (ω)∣≠0 with D≠0. The literature covers the first two forms thoroughly but the third form has been almost totally ignored. Therefore, it is now examined thoroughly, principally by analytical studies of simple bar structures and also by confirmatory numerical results for a rigidly jointed plane frame. Although structures are unlikely to have exactly the properties giving this form, it needs to be understood, particularly because ill-conditioning can occur for structures approximating those for which it occurs.